An Introduction
toThe
Gauge Integralalso known as
the generalized Riemann integral, the Henstock integral,
the Kurzweil integral, the Henstock-Kurzweil integral,
the HK-integral, the Denjoy-Perron integral, etc.

Historical and Bibliographical Overview

Integrals and derivatives were already known before Newton and Leibniz.
Those two mathematicians are generally credited with inventing
Calculus around 1670 because they developed its Fundamental
Theorem -- i.e., that areas are essentially the same
thing as antiderivatives. Later, Cauchy
investigated the integrals of continuous
functions.

Still later, Riemann refined the definition
that Cauchy had been using, and investigated the integrals of
discontinuous functions. The Riemann integral is simpler to
define than any of the other integrals discussed below, and it is the
"standard" integral that we teach to undergraduate students.
Still, it is more complicated than the antiderivative, and it seems to
be beyond the understanding of many undergraduate students.

The theory of the Riemann integral was not fully satisfactory. Many
important functions do not have a Riemann integral -- even after we
extend the class of integrable functions slightly by allowing
"improper" Riemann integrals. Moreover, even for integrable functions,
it is difficult to prove good convergence theorems using only the tools
ordinarily associated with Riemann integrals. A pointwise, bounded
limit of Riemann integrable functions is not necessarily Riemann
integrable. (For instance, since the rationals are enumerable, the
characteristic function of the rationals can be represented as the
pointwise limit of a sequence of characteristic functions of finite
sets.)

In 1902, Henri Lebesgue devised a new approach to integration,
overcoming many of the defects of the Riemann integral. Lebesgue's
definition is appreciably more complicated, but Lebesgue's techniques
yield better convergence theorems and, for the most part, more
integrable functions. The Lebesgue integral has become the
"standard" integral in our graduate courses in analysis.

The Lebesgue integral is strictly more general
than the proper Riemann integral -- i.e., it can integrate
a wider class of functions. However, in comparing the
improper Riemann integral with the Lebesgue integral, we find
that neither is strictly more general than the other.
Examples will be given later in this article.

Neither the improper Riemann integral nor the Lebesgue integral yielded
a fully satisfactory construction of antiderivatives. Slightly more
satisfactory answers -- i.e., more general notions of integral -- were
given by Arnaud Denjoy (1912) and Oskar Perron (1914). Denjoy's and
Perron's definitions turned out to be equivalent; both were very
complicated.

Decades later, independently, Ralph Henstock (1955) and Jaroslav
Kurzweil (1957) found a much simpler formulation of the Denjoy-Perron
integral. In fact, the Henstock-Kurzweil formulation -- the gauge
integral -- is considerably simpler than the Lebesgue idea, and its
definition is only slightly different from the definition of the
Riemann integral. Consequently, interest in this integral has been
rising over the last few decades, and some mathematicians have even
advocated that we should teach the gauge integral either alongside or
in place of either the Riemann integral or the Lebesgue integral.
I will express my own opinions about that possibility, at the
bottom of this web page. (In January 1997 I circulated
a letter advocating the use of the gauge
integral. It was signed by Henstock, Kurzweil, and several
other leaders of the gauge integral movement. I distributed it
to several publishers of calculus textbooks. However, it apparently
hasn't had any effect.)

Henstock, Kurzweil, and other early researchers in this area were
writing in rather specialized and technical terms for advanced
audiences, so their ideas did not spread quickly at first.
Here is a partial bibliography:

Mawhin, Introduction à l'analyse
2nd edition (1981). Contains some material on the gauge
integral.
(I don't know whether that was also true of the first edition.)

Muldowney, General theory of integration in function spaces,
including Wiener and Feynman integration, 1987.

DePree and Swartz, Introduction to Real Analysis, 1988, was
intended for advanced undergraduate students. It includes two
chapters of very readable material on the gauge integral.

Lanzhou Lectures on Henstock Integration by Lee Peng-Yee,
1989, a rather clear exposition of research-level material.

The integrals
of Lebesgue, Denjoy, Perron, and Henstock by Russell Gordon,
1994. This book is a comprehensive introduction to the subject.

Mawhin, Analyse, first edition (1992); 2nd
edition (1997).

Bartle's article,
"Return to the Riemann integral," in the October 1996
American Mathematical Monthly, is brief and easy to read;
probably it was read widely.

My book, published in October 1996, was
intended for beginning graduate students; I hope it is helpful. It
includes a chapter on the gauge integral, plus parts of several other
chapters.

A substantial number of articles in Real Analysis Exchange
are devoted to new discoveries concerning various aspects of the
gauge integral. These articles make up perhaps 5 or 10 percent of
the journal's content. This is the journal to read, for
anyone who wants to do research in this area. Of course, most of these
are research articles, not introductory articles; they will be unreadable
to anyone who does not already know the basics of the subject. --
I might mention in particular that
Gordon's article in Real Analysis Exchange22 (1) 1996/7,
pp. 23-33, gives a historical overview of this subject in much more
detail than the present web page.

Gordon's article in American Mathematical Monthly,
February 1998, discusses tagged partitions and gauges in interesting ways.

Robert G. Bartle and Donald R. Sherbert,
Introduction to Real Analysis, Third Edition,
Wiley, 2000. Suitable for advanced undergraduates.
Includes a chapter on the gauge integral.

Peng Yee Lee and Rudolf Vyborny,
The Integral:
An Easy Approach after Kurzweil
and Henstock,
Cambridge University Press, 2000. A combined
textbook/research
monograph, suitable for advanced undergraduates or beginning
graduate students. Very up-to-date -- for instance, it
includes
Bartle's recent theorem on negligible variation.

Jaroslav Kurzweil,
Henstock-Kurzweil Integration: Its Relation
to Topological Vector Spaces.
World Scientific Pub Co., 2000.
This
book is intended only for advanced specialists
in integration theory, not for a general audience.
It investigates topologies on the vector space
of all gauge-integrable functions.

Robert Bartle,
A Modern Theory of Integration,
Graduate Studies in Mathematics, AMS,
2001.
Suitable for advanced undergraduates or beginning
graduate students.

The Definition

To put the gauge integral into proper perspective, we should first
review the definition of the Riemann integral. For the moment, we
consider only the proper Riemann integral; improper integrals
will be discussed later. There are several different equivalent ways to
define the Riemann integral; among them, this one is best suited for
our present purposes:

Now a very slight modification yields the definition of the gauge
integral. Again, there are several different, equivalent ways to
define the gauge integral; we shall give only the formulation which
emphasizes the similarity between the Riemann and gauge integrals.

The only change is in the use of delta. Instead
of a positive constant, delta must be a
positive-valued function. The function delta is
called a gauge. It doesn't have to be continuous, or
measurable, or anything else; it just has to take strictly positive
values. Also note that

This
little change in the definition has enormous consequences in the
applications.

The collection of numbers n, si, ti
is called a tagged division (or a tagged
partition); the numbers si are called
the tags. The tagged division is called
δ-fine if it satisfies
the condition ti-ti-1 <
δ(si) for all i.
Some of the books and papers on this subject use
a slightly different condition, e.g., that
si-δ(si)
< ti-1 < ti <
si+δ(si).
This affects the development of the theory only
superficially -- e.g., in some places we need to
replace an ε with either
2ε or
ε/2.

Cousin's Lemma states that for any gauge
δ, there exists at least
one δ-fine tagged division.
Cousin's Lemma is not obvious; that is
admittedly a drawback in the gauge theory.
The proof of Cousin's Lemma uses the
fact that the
real number system has the Least Upper Bound property.

There are several slightly different ways to prove
Cousin's Lemma; here is a sketch of one of the proofs.
Let S={x∈[a,b] :
there exists a δ-fine tagged
division on [a,x]}. Then S is nonempty, since
a∈S.
Also, S is bounded above by b.
By the Least Upper Bound Property, the set S
has a supremum, or least upper bound; let us denote
that number by
σ. Then
σ∈[a,b].
If σ < b, use the
fact that
δ(σ)
>
0 to arrive at a contradiction. If
σ = b, use the fact that
δ(b) > 0 to prove that
the supremum is actually a maximum -- i.e., that
b is a member of S.

The set of all gauge-integrable functions from [a,b]
to R is a vector space -- i.e.,
any constant multiple of such a function or any
sum of two such functions is another such function.
That vector space is often known as the
Denjoy space, because of some work Denjoy
did in studying it.

We have only considered the simplest possible versions of the
definitions, but we remark that some generalizations are possible.

With slightly more complicated definitions, we could allow
plus and minus infinity as possible values
for the integrals. (Just equip the extended real line
with its usual convergence structure, and use
convergences instead of epsilons. This variant is used in the appendix
to Anatole Beck's book, Continuous Flows in the Plane.)

Or, instead of working with real-valued functions f, we could
work with functions f taking values in any Banach space X; most of the main
properties generalize easily to this setting. (See my book, for instance.)

We could also replace [a,b] with
other domains, but that is harder; it is discussed below.

Henstock and Kurzweil have also considered replacing the
expression f(si)(ti
ti1) with a more general expression g(si,
[ti1,ti]), where g is a
function of a real variable and an interval variable; but the results
in that case are more complicated and will not be considered
here.

Simplicity and Concreteness of the Definitions

The gauge integral is far simpler to define than the Lebesgue integral
-- it does not need to be preceded by explanations of sigma-algebras
and measures. Its simplicity stems from the fact that it makes good
use of the special properties of the interval [a,b], properties that
are not shared by all measure spaces. The basic proofs in the gauge
theory are very concrete and intuitive; we can give particular
examples of the functions delta. A simple
example is given in the next section of this article.

However, the fact that we're using properties of intervals means that
the gauge integral does not generalize readily to settings other than
intervals. It can be generalized to functions defined on the whole
real line, or to functions defined on finite-dimensional spaces, but
its definition and theory then become slightly more complicated. It
apparently can also be generalized to infinite dimensional spaces,
and more abstract setttings, but
I'm not very familiar with this theory, and I believe it's a lot more
complicated. Some of Henstock's writing deals with this theory.
I believe that Muldowney's book gives a more elementary treatment
of this theory, but I have to confess that I'm not very familiar
with the theory or with his book.

Contrast that with the Lebesgue approach, which requires far more
prerequisite equipment (sigma-algebras, measures, measurable sets,
measurable functions) but then generalizes without any difficulty at all
to settings entirely unrelated to intervals. Thus, the Lebesgue approach
may be preferable if one wants to study stochastic processes such as
Brownian motion. (Brownian motion can be explained in terms of
Wiener measure, a measure on an infinite-dimensional space of
continuous functions.)

Here is an example of the concreteness of the gauge integral: Any
introductory course on measure and integration should include a
construction of Lebesgue measure, but how should that be done? The
usual constructions are rather complicated and abstract; they have
little to do with the uses of Lebesgue measure. But it is
easy to define the gauge integral, and after we've defined it we can
proceed this way: Define

where 1S is the characteristic function of the set S. Then
the Lebesgue measurable subsets of [a,b] are those sets S for which
this gauge integral exists, and mu gives the
Lebesgue measure of such a set. This definition is extremely simple
and intuitive. Of course, one still has to do a fair amount of work
to prove that the Lebesgue measurable sets form a sigma-algebra and
the Lebesgue measure is countably additive on those sets.

A Simple Example

Generality: Existence of Integrals

In this discussion we consider only integrals of functions from [a,b]
to R. The following Venn diagram indicates the relations
between the most basic kinds of integrals. For instance,
every Lebesgue integrable function is also gauge integrable.
In other words, L1[a,b] is a subset of the
Denjoy space.

As indicated by the Venn diagram above, not every Lebesgue integral can
be viewed as a Riemann integral, or even as an improper Riemann
integral. For a simple example, consider the characteristic function
of the rationals.

As indicated by the Venn diagram above, not every improper Riemann
integral is a Lebesgue integral. For an example, let f(t) =
t2 cos(1/t2) with f(0)=0. Then
01|f '(t)|dt
is infinite, and the Lebesgue integral
01f '(t)dt
does not exist, but the improper Riemann integral
0Tf '(t)dt exists (for
any T) and equals f(T).

The examples above are typical. If g(t) is any gauge
integrable function on [a,b], then g(t) is Lebesgue
measurable. Hence the Lebesgue integral ab|g(t)|dt exists, if we
permit infinity as a possible value. The Lebesgue
integral abg(t)dt exists (with
its usual definition) if and only if ab|g(t)|dt is
finite. For any nonnegative function, the Lebesgue and
gauge integrals are the same.

This analogy may be helpful: The gauge integrable functions
are like convergent series; then the Lebesgue integrable
functions are like absolutely convergent series.
The absolutely convergent series are easier to work with,
and yield a tidier theory. The convergent series are
more general, but only occasionally do we need to work with
a series that is conditionally convergent.

Rapidly oscillating functions such as the derivative of t2
cos(1/t2)
are the only ones added to our collection of integrable functions when
we switch from the Lebesgue integral to the gauge integral. These
functions are more often used for pathological
counterexamples than for positive applications. Thus the
greater generality of the gauge theory does not make it
much more applicable. (A possible exception is in the
theory of differential equations; there seem to be some
important new applications there.) For the most part,
the big advantage of the gauge integral is the new
insight that it yields into the Lebesgue integral.

Here is another drawback to the gauge theory: The Lebesgue theory
yields the Banach space L1[a,b] of all Lebesgue-integrable
functions. It's a very nice space, with a nice metric and nice
convergence theorems. The gauge theory yields a slightly larger vector
space, which we might call G[a,b]. But the metric and the convergence
theorems for this space are much less nice. That's not surprising:
this space contains not only the nice functions of L1[a,b],
but also some additional functions that are rather nasty. If we don't
really need to work with those nasty functions, we might prefer to
stick to the setting of L1[a,b], which has nicer worktools
(i.e., the metric, the convergence theorems, etc.).

Fundamental Theorems

The gauge integral simplifies and strengthens the Fundamental Theorems
of Calculus:

the derivative F '(t) exists
and equals f(t), for almost every t in [a,b].

(b)

the derivative F '(t) exists and
equals f(t), at each t where f is continuous.

Result (a), or one like it, is what we teach our graduate students,
under the stronger assumption that f is Lebesgue integrable. Result
(b), or one like it, is what we teach our undergraduates -- usually
under the assumption that f is continuous throughout all of
[a,b], but that stronger assumption is not really needed.

Second
Fundamental Theorem of Calculus
(integrals of derivatives). Let F be a real-valued,
differentiable function on [a,b]. Then the gauge integral
ab F '(t)dt
exists
and equals F(b)F(a).

We teach our undergraduates a theorem like this, but it
usually has the additional hypothesis that the Riemann integral
ab F '(t)dt
exists. That is a conclusion, rather than a
hypothesis,
if we use the gauge integral.

Actually, the hypotheses can be made even weaker: We can
allow
F to be nondifferentiable at countably many points. More
precisely: Assume F is continuous, and that G is some
function
defined at every point of [a,b], and that F'(t) exists
and
equals G(t) for all but at most countably many values
of t. Then the gauge integral
ab G(t)dt
exists
and equals F(b)F(a).

Improper Integrals

The proper Riemann integral was not adequate for some elementary
purposes, so we extended it to an improper Riemann integral. For
instance, in our undergraduate courses,

But for the gauge integral, such an
extension is not needed; that is the content of

Thus, any "improper gauge integral" also exists as a proper
one.

(Clarification: The function f must be defined at
every point of [a,b]. In the case of
f(t)=1/√t on the
interval (0,1], we also assign some
value to f(0). What value we assign doesn't really
matter, just so long as f(0) is some particular number.)

Convergence Theorems

The theorem we usually teach to our undergraduates is this one:

Suppose (fn) is a
sequence of Riemann integrable functions on [a,b], converging uniformly
to a limit f. Then f and |fnf| are Riemann integrable, and
as well.

But uniform convergence is a very strong assumption, so we usually
teach our graduate students a result like this:

Dominated Convergence
Theorem. Suppose (fn) is a sequence of gauge integrable
functions, converging pointwise to a limit f. Also suppose that e(t)
< fn(t) < g(t) for
all n and t, where e and g are some gauge-integrable functions. Then
f and |fnf| are
gauge integrable, and as well.

We usually teach this to our graduate students under the additional
assumption that the functions e and g are Lebesgue integrable; then
the other functions are Lebesgue integrable too. However, the version
of the theorem presented above can be stated (without proof) in
undergraduate courses -- perhaps even in a freshman calculus course.

There are other convergence theorems for gauge integrals, involving
hypotheses weaker than the domination condition e(t)
< fn(t) < g(t),
but those theorems are generally much more complicated. Research is
still being done in that area.

The role of the gauge
integral in teaching analysis

The gauge integral is simple to define, and very concrete. Hence it
offers an improved intuition about integration; it improves our
understanding of the Lebesgue integral.

Here are my opinions (substantially revised 11/16/00).
The role of the gauge integral in our teaching must be
viewed in terms of the way that we presently teach
analysis. Here in the USA, we usually teach analysis on three levels:

The freshman calculus course, which uses the Riemann
integral and omits most proofs.

An advanced undergraduate course, typically using a
textbook titled "Introduction to Real Analysis" or
something like that. This course generally teaches how
to do analysis proofs; it includes proofs of the results
that were stated without proof in calculus.

A beginning graduate course, which generally covers
the Lebesgue integral.

Now, I think there is not much point in trying to
introduce the gauge integral on level (1). Our
freshman calculus students
have little or no understanding of proofs;
they concentrate on formulas and computations.
For instance, for most of those students, the
Second Fundamental Theorem of Calculus is simply the
equation
ab F '(t)dt = F(b)F(a);
any further clauses about continuity or differentiability
or existence of integrals are unnoticed by most students.
We show our calculus students that the characteristic
function of the rationals is not Riemann integrable
on (for instance) the interval [0,1], but
most of our calculus students have no idea what we are
talking about.

Level (2), on the other hand, would be an excellent place
to introduce the gauge integral. The conventional course
at the advanced undergraduate level concentrates on the
use of tools such as epsilons and deltas, convergent
sequences, limsups and liminfs, etc. Those are the same
tools used in the theory of the gauge integral. Thus,
adding that integral to the course would only involve
slight modifications in the course. The book by
DePree and Swartz is particularly well suited for this
purpose. Bartle and Sherbert, 3rd edition, is also
applicable to this purpose, though it devotes
slightly fewer pages to the gauge integral.

Introducing the gauge integral into
level (3), the graduate course, might also be a good
idea, but it would involve greater complication. The
graduate course has its share of epsilons and limsups,
but its most basic tools are
topologies and sigma-algebras, a very different sort
of thinking. The gauge integral does not generalize
readily to abstract settings like abstract
sigma-algebras. (It can be generalized to a setting
something like that, but it then loses much of
its wonderful simplicity.) In its simplest
form, it is more like the Lebesgue integral
with respect to Lebesgue measure, not other
measures. Of course, Lebesgue measure is
the most important measure. (Perhaps we
should deemphasize the other measures in
the first graduate course on integration
theory, in order to make the gauge integral
fit into the course more readily. Other
measures can be developed in greater
detail in a later course.)

I think that the two theories -- Lebesgue and
gauge -- both are valuable, and both deserve to
be covered in the graduate course. I'm not yet
sure what is the best way to organize such a
course. We can develop one of the two theories,
and then use its results as tools in developing
the other theory. This is the approach taken
by Gordon's book (which develops the Lebesgue
theory first) and by Bartle's "MTI" book and
the Lee/Vyborny book (both of which develop
the gauge theory first).

Some remarks on the lack of pathological examples

Examples are important, both in teaching and in research.
When we study any sort of integral, we prove that
certain classes of functions are integrable, and we give examples.
We also state that not all functions are integrable, and we
may perhaps give examples of that too. That
may be difficult, but it is important to do so, lest some
students get the impression that all functions are integrable.

Of course, there are some obvious
examples -- e.g., if we permit the function to be unbounded
or if we permit its domain to be unbounded. But what
about a bounded function on a bounded interval?
Within this class of functions, it is hard to give
an example of a function that is not integrable.
It was already hard enough with the Riemann integral -- for that
integral we had to use rather bizarre functions, such as the
characteristic function of the rationals. Now, when we turn to the
gauge integral or the Lebesgue integral, more
functions are integrable, and so it is even harder
to produce examples of non-integrable functions.

Actually, the problem is the same for gauge integrals or Lebesgue
integrals. Indeed, when f is a bounded real-valued function on a bounded
interval, then these three conditions are equivalent:

f is gauge integrable;

f is Lebesgue integrable;

f is Lebesgue measurable.

Thus, the problem of giving examples of nonintegrable functions is not
really a new one. But perhaps we are putting that problem in a new
context: We have suggested teaching the gauge integral to undergraduate
students, but it will be difficult to explain to undergraduates
that it is not true that

"every bounded function on a bounded interval is gauge
integrable."

Indeed, there do exist functions which contradict that
conjecture, but it is extremely difficult to
exhibit such a function.

Every introductory textbook on Lebesgue integrals includes a short
proof (due to Vitali) of the existence of a nonmeasurable set;
the characteristic function of that set is then a
nonmeasurable function. But
that proof, like every known proof of that theorem, is
nonconstructive -- it uses the Axiom
of Choice to prove the
existence of the nonmeasurable function without actually "finding" the
function or describing it explicitly. Vitali's proof is one of
the most elementary uses of the Axiom of Choice, and perhaps
it makes a good introduction to the Axiom of Choice; it could
be included in an appendix in a book intended for some advanced
undergraduate students. But I think this is conceptually
way beyond the
grasp of freshman calculus students. All we can do is tell these
students: "Yes, there does exist a non-integrable, bounded function on
a bounded interval, but describing it to you would require a great deal
of higher math, far beyond the scope of this freshman calculus course."

You might notice that, in the last paragraph, I said that every
known proof requires the Axiom of Choice.
Is it possible that we may someday find another
proof that is constructive? i.e., that we may find an explicit example
of an unmeasurable function? Probably not. This is not just a personal
opinion -- it has some supporting evidence, though the evidence is rather
technical. Solovay proved in 1970
that the nonexistence of a nonmeasurable function is consistent with ZF
set theory (i.e., conventional set theory minus the Axiom of Choice),
provided that there exists an inaccessible cardinal. The existence of
an inaccessible cardinal is empirically consistent with conventional
set theory -- i.e., after many decades, no one has yet found
a contradiction, so Solovay's model seems to be reliable.
If you're interested, these topics are discussed a little more in
my book, and more references are given there. --
My, we certainly seem to have gotten into some deep math, while
discussing what to teach to freshmen!

Some Open Research Problems

I'm not a leading researcher in this area, but I have
corresponded with several of the leaders. I asked them what
are some of the main problems still being researched. Here
are some of their responses:

Give a descriptive definition of the
Riemann integral. Saks mentioned in his
book that you can approach an integral
descriptively (like Newton -- e.g., the
integral is some sort of antiderivative) or
constructively (like Riemann -- e.g., the
integral is some sort of limit of approximating
sums). Many results
we have proved for the gauge integral are
not applicable to the Riemann integral, and the
above-mentioned question is one of them.

Lee Peng-Yee has given a Riemann-type definition
of the Ito integral using the Henstock
method. It definitely includes the Ito integral but
Lee Peng-Yee has not yet been able to determine
whether it also includes anything else -- i.e., is
it equivalent to the Ito integral, or is it more general?

Is there a natural topology for the
Denjoy space?
The sup norm gives us a natural topology on
C[0,1], and the norm
|| · ||1 gives us
a natural topology for L1[0,1]. Is there a
topology that is somehow natural for the Denjoy space (i.e., the
space of all gauge-integrable functions)? We do know
many convergence theorems for the gauge integral, as
mentioned earlier on this web page, but so far they have
not provided a good answer to this question about topologies.
Several topologies for the Denjoy space have been
proposed in the literature, but none of them seems
to have quite enough properties for us to call it
a "natural" topology. Some properties that would
be pleasant would be: metrizable;
complete or barrelled or
ultrabarreled; locally convex; and corresponding
in some natural way to some of the best known
convergence theorems about gauge integrals.
We may be able to get some of those properties,
but apparently not all of them; that is
made evident by discoveries in
Kurzweil's recent book on this subject.